I was reading this article ON Karaite jews about "Who is a Jew" it says:
However, anyone who formally accepts the God of Israel as his own god, the people of Israel as his own people, and is circumcised (males only), is a fully established member of the people of Israel (Jew); Karaites believe this should be done, after living amongst Karaites and studying the Torah, in the form of a vow before the Beit Din, see Exodus 12:43–49, Ruth 1:16, Esther 8:17, and Isaiah 56:6–7; also Ezekiel the prophet states that strangers who have joined themselves to the Children of Israel will be given land inheritance among the Tribes of Israel amongst whom they live during the final redemption.
Ezekiel 47:21-2317 וְחִלַּקְתֶּם אֶת-הָאָרֶץ הַזֹּאת לָכֶם—לְשִׁבְטֵי יִשְׂרָאֵל. וְהָיָה תַּפִּלוּ אוֹתָהּ בְּנַחֲלָה לָכֶם וּלְהַגֵּרִים הַגָּרִים בְּתוֹכְכֶם אֲשֶׁר-הוֹלִדוּ בָנִים בְּתוֹכְכֶם וְהָיוּ לָכֶם כְּאֶזְרָח בִּבְנֵי יִשְׂרָאֵל אִתְּכֶם יִפְּלוּ בְנַחֲלָה בְּתוֹךְ שִׁבְטֵי יִשְׂרָאֵל. וְהָיָה בַשֵּׁבֶט אֲשֶׁר-גָּר הַגֵּר אִתּוֹ שָׁם תִּתְּנוּ נַחֲלָתוֹ נְאֻם אֲדֹנָי יהוה So shall you divide this land unto you according to the tribes of Israel. And it shall be that you divide it by lot as an inheritance for yourselves, and for the resident aliens who sojourn in your midst and who bear children among you. And they shall be to you as native-born among the children of Israel—with you they have an inheritance in the midst of the tribes of Israel. And it shall be that in whatever tribe the resident alien sojourns, there you shall give him his inheritance," declares the Lord YHWH.
Although Muslims would never like to be called Jews not because they hate Bani Israel but because they love Bni Israel as they were the Muslims of the time and had never called themselves as Jew. but I found that beliefs of Karite Jews would cause Muslims to be recognized by the Karaite Judaism. I wanted an answer from a Karaite point of view on this issue. Naturally , the Kariate jews and Orthodox Jews would claim each other to be heretic or non Jew , hence i wanted this answer only from Karaite point of view.